Question

1. the profit, y, (in thousands of dollars) of a small business over time, x, (in months) can be modeled by a square root function. the business earned a profit for the first time on the second month, and the profit was $3 thousand dollars. by the 11th month, the profit was $4 thousand. write an equation to model this situation:

Explanation:

Step1: Identify the function type

The problem states the profit \( y \) (in thousands of dollars) over time \( x \) (in months) is a square root function. A general square root function is \( y = a\sqrt{x}+b \). But since it's the first profit on the 2nd month (maybe \( x = 2 \) when \( y = 0 \)? Wait, no, the first profit is on the second month, and profit is \( y \) (thousands). Wait, the first profit (maybe \( y = 0 \) before, and starts at \( x = 2 \))? Wait, no, let's re - read: "the business earned a profit for the first time on the second month, and the profit was $3 thousand dollars." So when \( x = 2 \), \( y = 3 \). And "By the 11th month, the profit was $4 thousand." So when \( x = 11 \), \( y = 4 \).

Wait, maybe the general form of the square root function is \( y=a\sqrt{x}+b \), but if we assume that at \( x = 2 \), \( y = 3 \) and at \( x = 11 \), \( y = 4 \). But maybe a simpler square root function: let's assume the form \( y = a\sqrt{x} \). Wait, when \( x = 2 \), \( y = 3 \): \( 3=a\sqrt{2}\), \( a=\frac{3}{\sqrt{2}} \), but then at \( x = 11 \), \( y=\frac{3}{\sqrt{2}}\sqrt{11}=\frac{3\sqrt{22}}{2}\approx\frac{3\times4.69}{2}\approx7.035\), which is not 4. So maybe the form is \( y = a\sqrt{x - c}+b \). Since the first profit is on the 2nd month, maybe \( c = 1 \) (so that at \( x = 2 \), \( x - c=1 \)), and \( b = 0 \). Let's try: when \( x = 2 \), \( y = 3 \): \( 3=a\sqrt{2 - 1}=a\times1\), so \( a = 3 \). Then check \( x = 11 \): \( y = 3\sqrt{11 - 1}=3\sqrt{10}\approx9.486\), not 4.

Wait, maybe the problem is that the profit function is \( y = a\sqrt{x} \), and we have two points: when \( x = 2 \), \( y = 3 \) and \( x = 11 \), \( y = 4 \). But that gives two equations:

1. \( 3=a\sqrt{2}\)
2. \( 4=a\sqrt{11}\)

If we solve for \( a \) from the first equation: \( a=\frac{3}{\sqrt{2}}\), from the second: \( a=\frac{4}{\sqrt{11}}\). These are not equal, so maybe the function is \( y = a\sqrt{x}+b \). Let's set up the system of equations:

When \( x = 2 \), \( y = 3 \): \( 3=a\sqrt{2}+b \)

When \( x = 11 \), \( y = 4 \): \( 4=a\sqrt{11}+b \)

Subtract the first equation from the second: \( 4 - 3=a(\sqrt{11}-\sqrt{2})\)

\( 1=a(\sqrt{11}-\sqrt{2})\)

\( a=\frac{1}{\sqrt{11}-\sqrt{2}}=\frac{\sqrt{11}+\sqrt{2}}{(\sqrt{11}-\sqrt{2})(\sqrt{11}+\sqrt{2})}=\frac{\sqrt{11}+\sqrt{2}}{11 - 2}=\frac{\sqrt{11}+\sqrt{2}}{9}\)

Then \( b = 3 - a\sqrt{2}=3-\frac{\sqrt{11}+\sqrt{2}}{9}\times\sqrt{2}=3-\frac{\sqrt{22}+2}{9}=\frac{27-\sqrt{22}-2}{9}=\frac{25 - \sqrt{22}}{9}\)

This seems complicated. Maybe the problem has a typo or we misinterpret. Wait, maybe the "first time on the second month" means that at \( x = 2 \), \( y = 3 \), and the function is \( y = k\sqrt{x} \), and we can also use the other point. Wait, maybe the intended form is \( y=a\sqrt{x} \), and we use two points to find \( a \), but since it's a square root function, maybe we can assume that the relationship is linear in terms of \( \sqrt{x} \). Let's let \( t=\sqrt{x} \), then \( y = mt + n \). When \( x = 2 \), \( t=\sqrt{2} \), \( y = 3 \); when \( x = 11 \), \( t=\sqrt{11} \), \( y = 4 \). The slope \( m=\frac{4 - 3}{\sqrt{11}-\sqrt{2}}=\frac{1}{\sqrt{11}-\sqrt{2}}=\frac{\sqrt{11}+\sqrt{2}}{9}\) as before. But maybe the problem is simpler, maybe the profit function is \( y=\sqrt{x + c} \). Let's try: when \( x = 2 \), \( y = 3 \): \( 3=\sqrt{2 + c}\), so \( 9=2 + c \), \( c = 7 \). Then check \( x = 11 \): \( y=\sqrt{11 + 7}=\sqrt{18}=3\sqrt{2}\approx4.24\), close to 4. Not exact.

Wait, maybe the problem is that the profit is a square root function of time, and we have two points: (2, 3) and (11, 4). Let's assume the function is \( y=a\sqrt{x}+b \). Let's plug in (2,3) and (11,4):

Equation 1: \( 3=a\sqrt{2}+b \)

Equation 2: \( 4=a\sqrt{11}+b \)

Subtract equation 1 from equation 2: \( 1=a(\sqrt{11}-\sqrt{2})\)

So \( a=\frac{1}{\sqrt{11}-\sqrt{2}}=\frac{\sqrt{11}+\sqrt{2}}{9}\approx\frac{3.3166 + 1.4142}{9}\approx\frac{4.7308}{9}\approx0.5256 \)

Then \( b = 3 - a\sqrt{2}\approx3-0.5256\times1.4142\approx3 - 0.743\approx2.257 \)

Then the equation is \( y=\frac{\sqrt{11}+\sqrt{2}}{9}\sqrt{x}+\frac{25 - \sqrt{22}}{9} \)

But this is messy. Maybe the problem intended to have integer values. Let's check if \( x = 2 \), \( y = 3 \): \( 3=\sqrt{9} \), so maybe \( x + 7=9 \), \( x = 2 \), so \( y=\sqrt{x + 7} \). Then \( x = 11 \), \( y=\sqrt{18}\approx4.24 \). Not 4. If \( x = 9 \), \( y = 4 \) (since \( \sqrt{9 + 7}=\sqrt{16}=4 \)), but the problem says 11th month. Maybe a typo, and it's the 9th month. Then the equation would be \( y=\sqrt{x + 7} \).

Alternatively, maybe the profit function is \( y = k\sqrt{x} \), and when \( x = 9 \), \( y = 3 \) (k = 1), but no. Wait, the problem says "the business earned a profit for the first time on the second month, and the profit was $3 thousand dollars. By the 11th month, the profit was $4 thousand."

Wait, maybe the function is \( y=\sqrt{x - 1} \). When \( x = 2 \), \( y=\sqrt{1}=1 \), no. \( y=\sqrt{x + 1} \), \( x = 2 \), \( y=\sqrt{3}\approx1.732 \), no.

Wait, maybe the general form of a square root function is \( y = a\sqrt{x} \), and we can use the two points to find \( a \). Let's set up a system:

For \( x = 2 \), \( y = 3 \): \( 3=a\sqrt{2}\) --> \( a=\frac{3}{\sqrt{2}}\)

For \( x = 11 \), \( y = 4 \): \( 4=a\sqrt{11}\) --> \( a=\frac{4}{\sqrt{11}}\)

Since \( \frac{3}{\sqrt{2}} eq\frac{4}{\sqrt{11}} \), the function is not of the form \( y = a\sqrt{x} \). So it must be of the form \( y=a\sqrt{x}+b \).

We have two equations:

1. \( 3=a\sqrt{2}+b \)
2. \( 4=a\sqrt{11}+b \)

Subtract equation 1 from equation 2:

\( 4 - 3=a(\sqrt{11}-\sqrt{2})\)

\( 1=a(\sqrt{11}-\sqrt{2})\)

\( a=\frac{1}{\sqrt{11}-\sqrt{2}}=\frac{\sqrt{11}+\sqrt{2}}{(\sqrt{11}-\sqrt{2})(\sqrt{11}+\sqrt{2})}=\frac{\sqrt{11}+\sqrt{2}}{9}\)

Then substitute \( a \) into equation 1:

\( 3=\frac{\sqrt{11}+\sqrt{2}}{9}\times\sqrt{2}+b \)

\( 3=\frac{\sqrt{22}+2}{9}+b \)

\( b = 3-\frac{\sqrt{22}+2}{9}=\frac{27-\sqrt{22}-2}{9}=\frac{25 - \sqrt{22}}{9}\)

So the equation is \( y=\frac{\sqrt{11}+\sqrt{2}}{9}\sqrt{x}+\frac{25 - \sqrt{22}}{9} \)

But this is a bit complicated. Maybe the problem expects a simpler approach, assuming that the profit function is \( y=\sqrt{x + c} \), and we made a mistake in the points. If we take \( x = 2 \), \( y = 3 \) (so \( 3=\sqrt{2 + c}\) --> \( c = 7 \)) and \( x = 9 \), \( y = 4 \) (since \( \sqrt{9 + 7}=\sqrt{16}=4 \)), maybe the 11th month is a typo and should be 9th month. Then the equation is \( y=\sqrt{x + 7} \)

Alternatively, if we consider that the profit is a square root function and the two points are (2,3) and (16,4) (since \( \sqrt{16}=4 \)), then the function is \( y=\sqrt{\frac{9}{2}x} \)? Wait, when \( x = 2 \), \( y=\sqrt{9}=3 \), when \( x = 16 \), \( y=\sqrt{72}=6\sqrt{2}\approx8.485 \), no.

Wait, maybe the problem is that the profit \( y \) (in thousands) and time \( x \) (in months) has a square root relationship, and we can write the equation using the two - point form for a linear function in terms of \( \sqrt{x} \). Let \( t = \sqrt{x} \), then \( y=mt + n \). We have two points \( (t_1,y_1)=(\sqrt{2},3) \) and \( (t_2,y_2)=(\sqrt{11},4) \). The slope \( m=\frac{y_2 - y_1}{t_2 - t_1}=\frac{4 - 3}{\sqrt{11}-\sqrt{2}}=\frac{1}{\sqrt{11}-\sqrt{2}}=\frac{\sqrt{11}+\sqrt{2}}{9} \)

Using point - slope form with \( (t_1,y_1) \):

\( y - 3=\frac{\sqrt{11}+\sqrt{2}}{9}(t - \sqrt{2}) \)

Substitute \( t=\sqrt{x} \):

\( y=\frac{\sqrt{11}+\sqrt{2}}{9}\sqrt{x}-\frac{\sqrt{11}+\sqrt{2}}{9}\times\sqrt{2}+3 \)

\( y=\frac{\sqrt{11}+\sqrt{2}}{9}\sqrt{x}-\frac{\sqrt{22}+2}{9}+3 \)

\( y=\frac{\sqrt{11}+\sqrt{2}}{9}\sqrt{x}+\frac{27-\sqrt{22}-2}{9} \)

\( y=\frac{\sqrt{11}+\sqrt{2}}{9}\sqrt{x}+\frac{25 - \sqrt{22}}{9} \)

Answer:

The equation of the square - root function modeling the profit \( y \) (in thousands of dollars) over time \( x \) (in months) is \( y=\frac{\sqrt{11}+\sqrt{2}}{9}\sqrt{x}+\frac{25 - \sqrt{22}}{9} \) (or a simplified form if there is a misinterpretation of the problem's data). If we assume a simpler form with a typo in the month value (e.g., 9th month instead of 11th month), the equation could be \( y = \sqrt{x + 7} \) when \( x = 2,y = 3 \) and \( x = 9,y = 4 \).